Shear Mechanical Properties of Rock Joints Under Non-Uniform Load Based on DEM

Abstract

Previous joint test studies have mainly been conducted under the condition of uniformly distributed loads, but in engineering, overlying loads are often non-uniformly distributed. It is necessary to investigate the shear mechanical properties of joints under non-uniform loads. This paper establishes three typical mechanical models: far-field concentration (FFC), near-field concentration (NFC), and focus on center (FOC). Direct shear test simulations using the DEM software PFC reveal that the location of load concentration affects main shear parameters. The closer the load concentration is to the far end along the shear direction, the greater the deformation of rock mass during failure is, the higher the proportion of shear cracks is, and the greater the strength and energy required for failure are. Compared to a uniformly distributed load (UD) condition, FOC shows similar mechanical properties; NFC yields inferior outcomes, while FFC results in superior mechanical performance compared to UD. By utilizing machine learning, five prediction models for non-uniform load shear strength are developed, with the Genetic-XGBoost algorithm demonstrating the highest accuracy. Weight calculation results indicate that load distribution form is the most critical factor influencing shear strength.

1. Introduction

Joints control the mechanical stability of rock masses [1], and their deterioration often leads to project destabilization, such as slope sliding and tunnel collapse, which seriously threaten life and property safety [2,3,4,5]. Typically, slip along the contact interface is the predominant mode of joint failure [6,7]. Therefore, investigating the shear mechanical properties of rock joints is imperative in computer-aided geotechnical engineering to understand potential hazards and formulate effective mitigation strategies.

The shear mechanical properties of joints are influenced by various factors including overburden thickness, roughness, filling conditions, and rock yield [8,9,10,11]. Numerous researchers have studied the impact of overburden loads on the shear mechanical properties of joints. For example, Esaki, Du [12] investigated the effect of shear deformation on the hydraulic conductivity of joints under normal load boundary conditions and found a similar trend between hydraulic conductivity and the shear expansion of joints. Xia, Yu [13] conducted shear tests on joints with different roughness with varying normal stiffness and infiltration pressure to reveal the hydraulic mechanism under normal stiffness (CNS) boundary conditions, considering effective normal stress interaction, normal stiffness, infiltration water pressure, and joint roughness. Zhang, Li [14] performed multi-stage creep tests on joints under different normal stresses to uncover the influencing mechanism of normal stress on creep characteristics. Liu, Liu [15] obtained factors influencing strength in jointed rock masses through a series of tests on structural surfaces under different circumferential pressures while proposing a criterion for strength under complex stress conditions. In terms of dynamics, Li, Gong [16] proposed two coupled loading methods, “critical static stress + slight disturbance” and “elastic static stress + impact disturbance”, to reveal the mechanical mechanism of rocks under dynamic loading. Okubo, Fukui [17] prepared tuff specimens under two conditions: natural air drying and saturation. Lajtai, Schmidtke [18] carried out uniaxial creep tests for 14.5 years to study the creep characteristics of rocks on a 10-year scale, analyzing the influence of water on rock creep mechanical properties based on indoor creep tests. Wu, Jiang [19] studied the shear properties of anchored joints under cyclic loading through laboratory shear tests, finding that the shear strength of bolted connections decreases more significantly than that of non-bolted connections.

The existing studies have effectively elucidated the mechanical mechanisms of joints under various conditions. However, these studies have been conducted based on the assumption of uniformly distributed overburden loads. In reality, due to diverse boundary conditions, overburden thicknesses along the joints often exhibit variations, resulting in a non-uniform distribution of overburden load (Figure 1). In this study, to investigate the mechanical properties of joints under non-uniform load and establish a shear theory that reflects their actual state, we initially constructed a non-uniform-load mechanical model incorporating common forms of overburden load distribution in engineering applications using the strip partitioning method. Subsequently, we conducted a series of simulations to examine the shear mechanical behavior of joints using the DEM software PFC while focusing on the impact different load distribution forms have on shear mechanical parameters, crack development, and energy evolution laws. Finally, we developed a shear strength criterion under non-uniform load using machine learning techniques, providing theoretical support for stability assessment and the reinforcement of rock-related projects.

2. Modeling of Non-Uniform Load Mechanics

Establishing a mechanics model that accurately reflects non-uniform load conditions is the foundation of this study. Assuming that the joint is smooth, we constructed an overburden load mechanics model using the strip division method [20]. According to this, three typical forms of overburden load distribution are observed on slopes and in tunnels (Figure 2). The first and second forms of load distribution are commonly encountered in tunneling conditions, where the load on the joint comes from the dead weight of the overlying strata (P = mgh), with its magnitude determined by the height of the overlying strata. Since the angle of the joint remains constant, both distribution forms (a) and (b) exhibit proportional increases or decreases in height along with shear direction while maintaining an isometric pattern. The difference L_d between adjacent loads depends on the angle α. The third type of load distribution, often found in slope conditions, features more overlying strata above the joint center that symmetrically decrease along both sides with a constant difference; thus, this results in greater overlying loads at the joint center tapering off evenly on both sides.

In this study, these three distribution forms are designated as far-field concentration (FFC), near-field concentration (NFC), and focus on central (FOC) based on their respective patterns along the shear direction indicating varying degrees of load concentration. To accurately represent how different modes of load distribution affect mechanical properties at joints, we divide each form into equal strips and use identical differences in applied loads for NFC and FFC distributions, respectively [21]. The magnitude distributions within NFC and FFC strips relative to the shear direction can be expressed as follows:

$$\left\{\begin{array}{l}{G}_T=N\cdot G_1+{\displaystyle \frac{N(N-1)G_d}{2}}\\ G_n=G_1+(n-1)G_d\end{array}\right.$$

(1)

where $G_T$ is the total overlying load, $N$ is the total number of strip loads, $G_n$ is non-uniform load at different positions, $n$ is the sequence number of each sub-load, $G_1$ is the first strip load along the shear direction, and $G_d$ is the load difference between each strip load.

For FOC, the load decreases with the same difference along the node, and the distribution of the load with the shear direction can be expressed as follows:

$$\left\{\begin{array}{l}{G}_T=2[{\displaystyle \frac{N}{2}}{G}_1+{\displaystyle \frac{\frac{N}{2}(\frac{N}{2}-1)G_d}{2}}]\\ G_n=G_1+(n-1)G_d\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(n\le {\displaystyle \frac{N}{2}})\\ G_n=G_{\frac{N}{2}}-(n-1)G_d\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(n>{\displaystyle \frac{N}{2}})\end{array}\right.$$

(2)

3. Numerical Modeling

3.1. Geometric Model

Current laboratory testing methods have limitations in achieving non-uniform loads. DEM is a numerical method based on modeling the behavior of granular materials, which is particularly suitable for studying the motion and interaction of particle systems under mechanical actions, such as rock mechanics. This study utilized the DEM software PFC 5.0 [22,23], which offers easily implemented boundary conditions and minimal dispersion, to address this issue. To accurately replicate the non-uniform load distribution found in natural settings, we selected a joint of natural rock that had sustained shear damage and conducted simulations based on contour lines obtained through scanning with a morphological instrument. This instrument, which comprised a set of platforms and devices, enabled rapid, non-contact 3D measurements, straightforward calibration, and robust analysis software. The coordinates of the contour lines were determined by measuring the position of the laser spot on the surface. Each point was scanned, resulting in corresponding x-, y-, and z-coordinate values with z representing the vertical height above the datum. The scanning resolution in the x-, y-, and z-directions was 0.5 μm. Data acquisition and analysis were automated using TalyMap 3.0 analysis software. The specific modeling steps were as follows: (1) Red sandstone underwent direct shear tests to obtain the structural surface of rock after shear damage. (2) The structural surface was scanned using an HL-3DC color 3D scanner to capture its 3D form (Figure 3b) and then processed with TalyMap software. (3) A 2D curve section along the shear direction was selected from the acquired structural surface contour and uniformly scaled to obtain a 100 mm-long section of contour data; these coordinate data were collected (Figure 3d). (4) The collected coordinates were imported into CAD software to generate a corresponding file in DXF format which was then imported into PFC for constructing a numerical geometric model of the rock body including a joint surface measuring with dimensions of 100 mm × 100 mm (Figure 3 illustrates the scanning process of the joint contour line).

3.2. Calibration of Microscopic Parameters

The microscopic parameters of the numerical model in this study were calibrated based on the agreement between the simulation and experimental results. However, sampling rock joints from field conditions and the inherent uncertainties in their properties pose challenges for laboratory direct shear tests. To replicate joint specimens for these tests [24], a mixture of water, high-strength cement, and fine quartz sand was used at a mass ratio of 1:2:4, following Xie, Lin [25]. Cement mortar has been widely employed as a model material to investigate the shear behavior of rock joints [26]. The size of the specimens was 100 mm × 100 mm. Details about specimen preparation can be found in reference [27], while Figure 4 and Figure 5 present the production flow chart and test results, respectively. Each group of tests was repeated three times under the same conditions, and the average value was taken as the curve data.

Considering both the mechanical properties of rocks and joints, values were assigned to them using a flat-joint model for rocks and smooth-joint model (in accordance with the mechanical characteristics of the joint surface) for joints [28]. The flat joint was used as a comparison object to calibrate micro-mechanical parameters for each contact model. The micro-parameter calibration results are presented in

Table 1. Calibration results of microscopic parameters.

Parameter Type	Microscopic Parameters	Value
Particle	Density (kg/m3)	2500
	Radius (mm)	0.6–0.9
	Porosity	0.1
Flat-joint contact model	Contact modulus, $\overline{E_c} \left(\mathrm{G}\mathrm{P}\mathrm{a}\right)$	1.8
	Contact stiffness ratio, $\overline{k_n}∕\overline{k_s}$	1.0
	Tensile strength (MPa)	13.8
	Cohesion (MPa)	28.8
	Internal friction angle (°)	32.5
	Friction coefficient, f	1.5
Smoot-joint contact model	Normal stiffness, $\mathrm{k}\mathrm{n} \left(\mathrm{G}\mathrm{P}\mathrm{a}\right)$	100
	Shear stiffness, $\mathrm{k}\mathrm{s} \left(\mathrm{G}\mathrm{P}\mathrm{a}\right)$	0.62
	Friction coefficient, f	0.92

.

3.3. Boundary Conditions and Loading Scheme

In this study, the top loading plate was divided into 10 smaller sections (N = 10), and distinct loads were applied to achieve a non-uniform distribution of overburden loads; as illustrated in Figure 6, there are different differences in the initial stage of the shear stress–shear displacement curve, that is, in the micro-crack compaction stage, which were caused by the difference in normal stress and the dispersion of the sample itself. For the FFC and NFC conditions, G_d was set at 1.1% G_T and −1.1% G_T, respectively. G₁ was set at 15% G_T for the FFC condition and 5% G_T for the NFC condition. According to Equation (1), the calculation formula for the load division of each strip is shown as follows:

$$G_n=\left\{\begin{array}{l}15\%\cdot G_T-1.1\%\cdot (n-1)G_T\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(NFC\right)\\ 5\%\cdot G_T+1.1\%\cdot (n-1)\times G_T\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(NFC\right)\end{array}\right.$$

(3)

For FOC, the strip load had a centrally symmetric isometric distribution with d set at 2.5% G_T, while G₅ and G₆ were both set at 15% G_T. According to Equation (2), the calculation formula for the strip load is as follows:

$$G_n=\left\{\begin{array}{l}5\%G_T+2.5\%(n-1)G_T\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n\le 5\\ 27.5\%G_T-2.5\%(n-1)G_T\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n>5\end{array}\right.$$

(4)

As part of simulating direct shear tests, four different forms of load distribution under various overburden loads (G_T = 10 kN, 15 kN, 20 kN, 25 kN, 30 kN) were designed, as shown in

Table 2. Experimental simulation scheme.

Distribution Form	$\mathbf{Total} \mathbf{Normal} \mathbf{Load} {\mathit{G}}_{\mathit{T}}$	Normal Stress σ
Uniform distribution (UD)	10 kN, 15 kN, 20 kN, 25 kN, 30 kN	1 MPa, 1.5 MPa, 2 MPa, 2.5 MPa, 3 MPa
Far-field concentration (FFC)
Near-field concentration (NFC)
Focus on center (FOC)

.

4. Simulation Results and Analysis

4.1. Shear Stress–Shear Displacement Curve

Upon completing the simulation of the direct shear test, the shear stress–displacement curves for the joint surface corresponding to three distinct load distribution forms and a uniform load distribution (UD) form were obtained, as shown in Figure 7. In the initial loading phase, peak shear stresses related to various load and distribution forms closely aligned with the slope of the curve. As loading progressed, there was a deviation between the slope of the curve, peak shear displacement, and shear strength; higher normal stress combined with a load concentration location closer to the distal end resulted in increased corresponding shear strength. The closer the load concentration location was to the far end of the shearing direction, the stronger the damage resistance of the rock mass structure was.

The relationship between normal stress and peak shear displacement is illustrated in Figure 8. Generally, peak shear displacements increased with increasing normal stress, although the average values varied across different load distribution forms. Specifically, the average peak shear displacements for the uniform distribution (UD), near-field concentration (NFC), focus-on-central (FOC), and far-field concentration (FFC) load distribution forms were 1.634 mm, 1.576 mm, 1.656 mm, and 1.728 mm, respectively. In comparison to the NFC condition, the average peak shear displacements for FOC and FFC increased by 13.72% and 18.64%, respectively. This indicates that the average peak shear displacement increased with the location of load concentration and could be well approximated by a linear expression, as shown in Equation (5). This finding emphasizes the significant influence of load distribution form on peak shear displacement.

$$\stackrel{-}{{\gamma }_{\mathrm{p}}}=0.0152{\gamma }_P+1.577$$

(5)

where $\stackrel{-}{{\gamma }_{\mathrm{p}}}$ is the average peak shear displacement and ${\gamma }_P$ is the peak shear displacement, both in mm.

To further analyze the influence of load distribution form on shear stiffness, the slope (shear stiffness) of the shear stress–shear displacement curve under normal stress 2.0 MPa was calculated as a change in the form of shear displacement, as shown in Figure 9. It can be seen from the figure that the shear stiffness in all distribution forms decreased with the increase in shear displacement, but the rate of decrease was more and more slow, which corresponded to the evolution of the elastic stage–damage-softening stage of the shear stress–shear displacement curve. However, according to the slope of the rigidity–shear displacement fitting curve under different load distribution forms, the attenuation trend of shear stiffness with shear displacement was k_NFC > k_FOC > k_FFC. In terms of average stiffness, the average values of k_NFC, k_FOC, and k_FFC were 2.95, 2.99, and 3.11, respectively, and k_FFC was 4.1% larger than k_FOC. It was 5.4% larger than k_FFC, indicating that the stiffness resistance to decay under FFC conditions was stronger and less prone to damage.

4.2. Main Shear Mechanical Parameters

The relationship between normal stress and shear strength is illustrated in Figure 10. Under the same normal load, the shear strength corresponding to FFC was the highest, followed by FOC, with the lowest for NFC. It should be noted that the shear strength under FOC conditions was more similar to that under UD conditions in terms of numerical magnitude and trend. Further analysis of shear strength was conducted using the Mohr–Coulomb criterion [29], as shown in Figure 11. The cohesion increased with an increase in the location of load concentration. Compared with the NFC condition, the cohesion increased by 12.4% under FOC and by 30.76% under FFC conditions, respectively. The internal friction angle decreased with an increase in the location of load concentration as well. Compared to the NFC condition, there was a decrease of 0.91% in the internal friction angle under FOC conditions and a decrease of 1.44% under FFC conditions.

4.3. Crack Evolution

To investigate the law of damage evolution, crack extensions for different forms of load distribution are presented in Figure 12 (G_T = 20 kN), at various stress levels (τ = 50% τ_max; τ = 70% τ_max; τ = 90% τ_max; τ = τ_max; τ = τ_failure). Shear cracks are represented by the red color, while the green color indicates tensile cracks. From observing crack propagation in different load distribution modes, it can be seen that the main cause of rock mass failure was micro-tensile cracking at the highest joint center. With an increase in shear stress level, cracks initiated at the highest undulation (the center of the joint surface) and gradually expanded along the lower slope to form two macro-through-cracks. It is worth noting that there was only one through-crack for NFC and FFC at the peak shear stress, while under UD and FFC conditions, the crack growth mode was essentially similar with two obvious through-cracks at the peak shear stress. The results indicate that compared to FOC and UD conditions, NFC and FFC conditions required more damage for rock mass failure and made it more difficult to destroy the rock mass structure.

Figure 13 delineates the spatial–temporal evolution of tensile versus shear fracture distribution across distinct loading phases under varying load configuration modes. During the initial loading stages (NFC, FOC, and UD conditions), tensile fractures prevailed as the principal crack type due to localized stress concentrations exceeding the material’s tensile resistance. With progressive loading, a characteristic transition occurred where shear fractures gradually dominated the failure process, achieving predominance at 90% τ_max under NFC conditions. This transition occurred earlier at 70% τ_max for FOC and UD configurations, indicating accelerated shear localization under these boundary constraints. Contrastingly, FFC conditions exhibited sustained shear fracture dominance throughout the entire loading history, confirming shear-driven failure as the fundamental fracture propagation mechanism under fully confined conditions.

The observed fracture pattern transition reveals two distinct failure pathways: (1) For the NFC, FOC, and UD conditions, initial tensile-dominated damage evolved into shear-controlled failure through progressive micro-crack coalescence and stress redistribution during shear deformation. (2) FFC conditions demonstrated immediate shear fracture nucleation and propagation due to enhanced confinement restricting tensile crack development. This mechanistic difference explains the enhanced shear strength under FFC loading; the energy-intensive shear fracture process (characterized by frictional sliding and asperity degradation) requires greater energy dissipation compared to tensile fracture mechanisms (governed by bond separation with lower energy consumption). The resultant strength hierarchy (FFC > NFC ≈ FOC) directly correlated with the proportion of high-energy shear fractures in the failure process.

4.4. Energy Evolution Law

The essence of the mechanical behavior of rocks under loading lies in the process of external energy input, storage, and dissipation. Examining the rock damage process from an energy perspective can effectively reveal the underlying mechanical mechanisms of rocks. Assuming that no energy exchange occurs between the rock loading system and the external environment, according to the law of the conservation of energy, the relationship between absorbed energy (U), elastic energy (U_e), and dissipated energy (U_d) generated by microdefect closure or expansion can be expressed as

$$U=U_e+U_d$$

(6)

where U is the energy absorbed in loading; U_e is the elastic energy stored inside the rock and released during the unloading process; U_d is the energy consumed during the plastic deformation of the rock, which is irreversible.

Equation (6) can be expressed in the integral form of energy density as

$$\int udV=\int u_{e}dV+\int u_{d}dV$$

(7)

where u represents the input energy density, U_e denotes the elastic energy density, U_d signifies the dissipative energy density, and dV represents the sum of the energy density corresponding to the unit volume.

These parameters could be calculated by integrating the curves [30], as shown in Figure 14. The lower stress point could be set to 30% of shear strength, and the area of trapezoid ABCF represents the input energy density for the loading process. Trapezoid ABCED indicates the elastic energy density that can be released. Trapezoid CDEF represents the dissipated energy density. The calculation of these areas can be obtained from Equation (8):

$$\left\{\begin{array}{l}{u}^n={\int }_{{\gamma }_n^n}^{{\gamma }_n^m}{\tau }_{max}d\gamma \\ u_e^n={\int }_{{\gamma }_n^m}^{{\gamma }_n^m}{\tau }_{low}d\gamma \\ u_d^n=u^n-u_e^n={\int }_{{\gamma }_n^n}^{{\gamma }_m^m}{\tau }_{max}d\gamma -{\int }_{{\gamma }_n^n}^{{\gamma }_n^m}{\tau }_{low}d\gamma \end{array}\right.,$$

(8)

where τ_max and τ_low are used to calculate the maximum and minimum shear stresses, respectively.

The evolution relationships of absorbed energy (U), elastic energy (U_e), and dissipated energy (U_d) are presented in Figure 15 to analyze the energy level of the joint surface under different load distribution methods. As the stress level increased, both U and U_e showed a gradual rise, except under UD conditions, while U_d did not display a discernible trend. The average values of U and U_e for FFC were larger than those for FOC and NFC, and the average values of U_d for FFC were greater than those for NFC and FOC. These differences stemmed from the varying degrees of crack evolution under different forms of load distribution; FFC was characterized by shear cracks with higher strength and greater energy dissipation, whereas NFC and FOC were dominated by tensile cracks with lower strength and less energy dissipation. In practical engineering, it is advisable to avoid placing load concentration locations too close together. Based on the above results and analysis, it can be observed that as the load concentration location becomes closer to the far end along the shear direction, there is greater deformation in rock mass during failure, requiring more strength and energy for failure. Increasing the distance between load concentration locations makes the rock mass structure less prone to damage; therefore, it should be avoided when load concentration locations are too close together but rather placed in positions with large load concentrations.

5. Non-Uniform-Load Shear Strength Criterion Based on Machine Learning Methods

The shear strength criterion provides an intuitive theoretical explanation for the strength properties of rock masses. However, due to the challenges in quantitatively describing the non-uniform distribution characteristics of loads, traditional shear strength criteria with quantitative parameters have become less applicable. In this study, prediction models were constructed using the XGBoost algorithm based on simulation results from direct shear tests. This algorithm can describe parameters both qualitatively and quantitatively. Additionally, XGBoost algorithms optimized by Whale, Particle Swarm, Genetics, and Bayesian algorithms were employed. By comparing the model’s predicted values with the test data, we selected the algorithm that exhibited higher accuracy to build our prediction model.

The XGBoost (Extreme Gradient Boosting) model is an integrated machine learning algorithm based on decision trees with a gradient boosting framework. XGBoost possesses numerous advantages, such as high accuracy, flexibility, impressive learning efficiency, and parallelizable algorithms, making it widely used in various industries for big data processing. The specific approach involves continuously splitting and growing the tree using predetermined feature values. Each additional tree learns a new function to fit the residuals of the previous prediction. Ultimately, training is completed by yielding k trees, and the score of a sample is predicted by combining these k trees. During this process, the CART regression tree model is employed. CART assumes that the regression tree is binary and performs splitting based on feature values; for instance, tree nodes are split according to the j-th feature value and divided into left and right subtrees depending on whether the feature value is less than s:

$$\left\{\begin{array}{l}{R}_1(j,s)=\left\{x\mid x^{(j)}\le s\right\} \\ R_2(j,s)=\left\{x\mid x^{(j)}>s\right\}\end{array}\right.$$

(9)

The typical objective function of a CART regression tree is

$$\sum _{x_i\in R_m}{\left(y_i-f\left(x_i\right)\right)}^2$$

(10)

When solving for the optimal cut-off feature j and the optimal cut-off point s, the function can be transformed as

$$\underset{j,s}{\min }\left[\underset{c_1}{\min }\sum _{x_i\in R_1\left(j,s\right)}{\left(y_i-c_1\right)}^2+\underset{c_2}{\min }\sum _{x_i\in R_2\left(j,s\right)}{\left(y_i-c_2\right)}^2\right]$$

(11)

With the above equations, we can obtain the optimal cut features and cut points and finally build the regression tree. When the training is completed to obtain k trees, the scores corresponding to each tree are accumulated to obtain the predicted value of that sample. The specific calculation equation is as follows:

$$\widehat{y}=\phi \left(x_i\right)=\sum _{k=1}^K{f}_k\left(x_i\right)$$

(12)

where

$$F=\left\{f\left(x\right)=w_{q\left(x\right)}\right\}\left(q:R^m\to T,w\in R^T\right)$$

(13)

where $w_{q(x)}$ is the fraction of leaf nodes and f(x) is one of the regression trees.

The dataset used to construct the strength prediction model consisted of test results from four different distribution forms under 21 different normal loads, totaling 84 sets of data (

Table 3. Sample dataset.

Normal Stress (MPa)	Shear Strength Simulation (MPa)
	NFC	FOC	FFC	UD
1	3.6	3.78	4.21	3.77
1.1	3.71	3.93	4.22	3.9
1.2	3.84	4.29	4.38	4.31
1.3	4.11	4.54	4.82	4.55
1.4	4.25	4.53	5.08	4.71
1.5	4.29	4.71	5.28	4.74
1.6	4.41	4.67	5.31	4.76
1.7	4.52	4.84	5.31	4.92
1.8	4.52	4.74	5.44	5.06
1.9	4.66	5.06	5.58	4.84
2	4.79	5.2	5.75	5.13
2.1	5.05	2.16	5.77	5.1
2.2	5.01	5.28	5.88	5.22
2.3	5.12	5.4	5.91	5.34
2.4	5.19	5.53	5.97	5.46
2.5	5.24	5.58	5.94	5.56
2.6	5.3	5.63	6.21	5.67
2.7	5.35	5.69	5.29	5.71
2.8	5.38	5.7	6.21	5.75
2.9	5.41	5.81	6.26	5.83
3	5.6	5.91	6.31	5.8

). It could be found that in these data, FFC had the highest strength under the same normal stress condition, which was consistent with the conclusion above. The model was trained using 75% of the samples as the training set, while the remaining 25% served as the test set for validation. Among all algorithms tested, Genetic-XGBoost yielded the highest R² value of 0.973 for predicting shear strength and was therefore chosen to construct the model. A weight algorithm module was introduced to calculate weights corresponding to load value and load distribution form [31]. The load distribution form had a significantly larger weight value (76.7%) than load value (23.2%), indicating that it is a more dominant factor affecting shear strength compared to uniform conditions where overburden load remains an important influencing factor. This study highlights that considering load distribution form is critical when predicting shear strength and ensuring rock mass stability.

6. Conclusions

In this study, we constructed a non-uniform load mechanical model using the strip division method to investigate the effect of non-uniform load on the shear mechanical properties of joint surfaces. We then conducted a series of shear test simulations based on PFC software to analyze how different load distribution forms affect main shear parameters (cohesion and internal friction angle), crack expansion patterns, and energy evolution laws. Finally, we used machine learning methods to develop a quantitative prediction model for shear strength.

(1) Taking into account the common overburden loads on joint surfaces in engineering projects, we established a non-uniform-load mechanics model for joint surfaces using the strip partitioning method. This resulted in three representative forms of load distribution: far-field concentration (FFC), near-field concentration (NFC), and focus on central (FOC). Additionally, based on idealized mathematical hypotheses, we derived load distribution formulas for these three conditions.

(2) The proximity of load concentration to the distal shear direction correlated positively with peak shear displacement and strength. Compared to NFC, cohesion increased by 12.4% (FOC) and 30.76% (FFC), whereas internal friction angles decreased marginally by 0.91% (FOC) and 1.44% (FFC). Fracture analysis revealed distinct failure modes: NFC and FOC specimens exhibited one through-crack at the peak shear stress, while FFC developed two. Initial loading stages in NFC/FOC prioritized tensile cracks, transitioning to shear-dominated failure. Notably, shear cracks comprised >50% of total crack propagation in FFC. Energy dissipation metrics showed hierarchical trends: deformation energy (U) and elastic energy (Ue) followed FFC > FOC > NFC, while dissipated energy (Ud) ranked FFC > NFC > FOC, reflecting crack evolution differences under varying loading configurations.

(3) According to the characteristics of deformation, strength, crack formation, and energy evolution under different load distribution forms, the farther the maximum load location was along the shear direction, the stronger the resistance to damage and stability of rock mass was. In engineering practice, it is advisable to position the maximum load as far away as possible from the joint along the shear direction.

(4) By using the XGBoost algorithm, we proposed five quantitative shear strength prediction models: the Whale, Genetic, Bayesian, and Particle Swarm algorithms. By comparing the predicted values with the test values, we obtained the Genetic-XGBoost algorithm prediction model with the highest prediction accuracy (R² = 0.973). The weight of the load distribution mode was 76.7%, which was considerably higher than that of the load value at 23.2%. This indicates that load distribution form is a primary factor affecting shear strength.